Multi Degree of Freedom Holonomic System¶
In this example we demonstrate the use of the functionality provided in
sympy.physics.mechanics
for deriving the equations of motion (EOM) of a
holonomic system that includes both particles and rigid bodies with contributing
forces and torques, some of which are specified forces and torques. The system
is shown below:
![../../../_images/multidof-holonomic.png](../../../_images/multidof-holonomic.png)
The system will be modeled using System
. First we need to create the
dynamicsymbols()
needed to describe the system as shown in the above
diagram. In this case, the generalized coordinates \(q_1\) represent lateral
distance of block from wall, \(q_2\) represents angle of the compound
pendulum from vertical, \(q_3\) represents angle of the simple pendulum
from the compound pendulum. The generalized speeds \(u_1\) represents
lateral speed of block, \(u_2\) represents lateral speed of compound
pendulum and \(u_3\) represents angular speed of C relative to B.
We also create some symbols()
to represent the length and mass of the
pendulum, as well as gravity and others.
>>> from sympy import zeros, symbols
>>> from sympy.physics.mechanics import *
>>> q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1, q2, q3, u1, u2, u3')
>>> F, T = dynamicsymbols('F, T')
>>> l, k, c, g, kT = symbols('l, k, c, g, kT')
>>> ma, mb, mc, IBzz= symbols('ma, mb, mc, IBzz')
With all symbols defined, we can now define the bodies and initialize our
instance of System
.
>>> wall = RigidBody('N')
>>> block = Particle('A', mass=ma)
>>> compound_pend = RigidBody('B', mass=mb)
>>> compound_pend.central_inertia = inertia(compound_pend.frame, 0, 0, IBzz)
>>> simple_pend = Particle('C', mass=mc)
>>> system = System.from_newtonian(wall)
>>> system.add_bodies(block, compound_pend, simple_pend)
Next, we connect the bodies using joints to establish the kinematics. Note that we specify the intermediate frames for both particles, as particles do not have an associated frame.
>>> block_frame = ReferenceFrame('A')
>>> block.masscenter.set_vel(block_frame, 0)
>>> slider = PrismaticJoint('J1', wall, block, coordinates=q1, speeds=u1,
... child_interframe=block_frame)
>>> rev1 = PinJoint('J2', block, compound_pend, coordinates=q2, speeds=u2,
... joint_axis=wall.z, child_point=l*2/3*compound_pend.y,
... parent_interframe=block_frame)
>>> simple_pend_frame = ReferenceFrame('C')
>>> simple_pend.masscenter.set_vel(simple_pend_frame, 0)
>>> rev2 = PinJoint('J3', compound_pend, simple_pend, coordinates=q3,
... speeds=u3, joint_axis=compound_pend.z,
... parent_point=-l/3*compound_pend.y,
... child_point=l*simple_pend_frame.y,
... child_interframe=simple_pend_frame)
>>> system.add_joints(slider, rev1, rev2)
Now we can apply loads (forces and torques) to the bodies, gravity acts on all bodies, a linear spring and damper act on block and wall, a rotational linear spring acts on C relative to B specified torque T acts on compound_pend and block, specified force F acts on block.
>>> system.apply_uniform_gravity(-g * wall.y)
>>> system.add_loads(Force(block, F * wall.x))
>>> spring_damper_path = LinearPathway(wall.masscenter, block.masscenter)
>>> system.add_actuators(
... LinearSpring(k, spring_damper_path),
... LinearDamper(c, spring_damper_path),
... TorqueActuator(T, wall.z, compound_pend, wall),
... TorqueActuator(kT * q3, wall.z, compound_pend, simple_pend_frame),
... )
With the system setup, we can now form the equations of motion with
KanesMethod
in the backend.
>>> system.form_eoms(explicit_kinematics=True)
Matrix([
[ -c*u1(t) - k*q1(t) + 2*l*mb*u2(t)**2*sin(q2(t))/3 - l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t)))*Derivative(u3(t), t) - l*mc*(-sin(q2(t))*cos(q3(t)) - sin(q3(t))*cos(q2(t)))*(u2(t) + u3(t))**2 + l*mc*u2(t)**2*sin(q2(t)) - (2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))))*Derivative(u2(t), t) - (ma + mb + mc)*Derivative(u1(t), t) + F(t)],
[-2*g*l*mb*sin(q2(t))/3 - g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - g*l*mc*sin(q2(t)) + l**2*mc*(u2(t) + u3(t))**2*sin(q3(t)) - l**2*mc*u2(t)**2*sin(q3(t)) - mc*(l**2*cos(q3(t)) + l**2)*Derivative(u3(t), t) - (2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))))*Derivative(u1(t), t) - (IBzz + 4*l**2*mb/9 + mc*(2*l**2*cos(q3(t)) + 2*l**2))*Derivative(u2(t), t) + T(t)],
[ -g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - kT*q3(t) - l**2*mc*u2(t)**2*sin(q3(t)) - l**2*mc*Derivative(u3(t), t) - l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t)))*Derivative(u1(t), t) - mc*(l**2*cos(q3(t)) + l**2)*Derivative(u2(t), t)]])
>>> system.mass_matrix_full
Matrix([
[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0],
[0, 0, 0, ma + mb + mc, 2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))), l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t)))],
[0, 0, 0, 2*l*mb*cos(q2(t))/3 + mc*(l*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l*cos(q2(t))), IBzz + 4*l**2*mb/9 + mc*(2*l**2*cos(q3(t)) + 2*l**2), mc*(l**2*cos(q3(t)) + l**2)],
[0, 0, 0, l*mc*(-sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))), mc*(l**2*cos(q3(t)) + l**2), l**2*mc]])
>>> system.forcing_full
Matrix([
[ u1(t)],
[ u2(t)],
[ u3(t)],
[ -c*u1(t) - k*q1(t) + 2*l*mb*u2(t)**2*sin(q2(t))/3 - l*mc*(-sin(q2(t))*cos(q3(t)) - sin(q3(t))*cos(q2(t)))*(u2(t) + u3(t))**2 + l*mc*u2(t)**2*sin(q2(t)) + F(t)],
[-2*g*l*mb*sin(q2(t))/3 - g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - g*l*mc*sin(q2(t)) + l**2*mc*(u2(t) + u3(t))**2*sin(q3(t)) - l**2*mc*u2(t)**2*sin(q3(t)) + T(t)],
[ -g*l*mc*(sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t))) - kT*q3(t) - l**2*mc*u2(t)**2*sin(q3(t))]])